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**A second order \(\eta \)-approximation method for constrained optimization problems involving second order invex functions.**
*(English)*
Zbl 1212.90307

Summary: A new approach for obtaining second order sufficient conditions for nonlinear mathematical programming problems which makes use of the second order derivative is presented. In the so-called second order \(\eta \)-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order \(\eta \)-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second order \(\eta \)-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.

### MSC:

90C26 | Nonconvex programming, global optimization |

90C30 | Nonlinear programming |

90C46 | Optimality conditions and duality in mathematical programming |

### Keywords:

second order \(\eta \)-approximated optimization problem; second order invex function; second order optimality conditions
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\textit{T. Antczak}, Appl. Math., Praha 54, No. 5, 433--445 (2009; Zbl 1212.90307)

### References:

[1] | T. Antczak: An {\(\eta\)}-approximation approach to nonlinear mathematical programming problems involving invex functions. Numer. Funct. Anal. Optimization 25 (2004), 423–438. · Zbl 1071.90032 |

[2] | M. S. Bazaraa, H. D. Sherali, C. M. Shetty: Nonlinear Programming. Theory and Algorithms. John Wiley & Sons, New York, 1993. |

[3] | C. R. Bector, B. K. Bector: (Generalized)-bonvex functions and second order duality for a nonlinear programming problem. Congr. Numerantium 52 (1985), 37–52. |

[4] | C. R. Bector, M. K. Bector: On various duality theorems for second order duality in nonlinear programming. Cah. Cent. Etud. Rech. Opér. 28 (1986), 283–292. · Zbl 0622.90068 |

[5] | C. R. Bector, S. Chandra: Generalized bonvex functions and second order duality in mathematical programming. Res. Rep. 85-2. Department of Actuarial and Management Sciences, University of Manitoba, Winnipeg, 1985. |

[6] | C. R. Bector, S. Chandra: (Generalized) bonvexity and higher order duality for fractional programming. Opsearch 24 (1987), 143–154. · Zbl 0638.90095 |

[7] | A. Ben-Israel, B. Mond: What is invexity? J. Aust. Math. Soc. Ser. B 28 (1986), 1–9. · Zbl 0603.90119 |

[8] | A. Ben-Tal: Second-order and related extremality conditions in nonlinear programming. J. Optimization Theory Appl. 31 (1980), 143–165. · Zbl 0416.90062 |

[9] | B. D. Craven: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24 (1981), 357–366. · Zbl 0452.90066 |

[10] | M. A. Hanson: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981), 545–550. · Zbl 0463.90080 |

[11] | O. L. Mangasarian: Nonlinear Programming. McGraw-Hill, New York, 1969. |

[12] | D. H. Martin: The essence of invexity. J. Optimization Theory Appl. 47 (1985), 65–76. · Zbl 0552.90077 |

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